Do flavor conservation and spontaneous CP nonconservation lead to a real Kobayashi-Maskawa matrix

We reexamine the implication of flavor conservation in tree-level neutral- Higgs-bosonexchange for multi-Higgs-scalar SU(2) x U(1) models of spontaneous CP nonconservation . Contrary to a previous claim, we show that in such models for an arbitrary number of fermion families the quark mixing matrix does not have to be real. However, the complex structure derived for the Kobayashi-Maskawa matrix in the three-family model is shown to be in conflict with experiment.